Is possible to save JavaScript variable as file?

You can use a data: URI like in adeneo's answer, but another way is to use an HTML5 Blob and createObjectURL (similarly using the download attribute to create a download link).

The benefit of using createObjectURL is that there are severe size limits to data URIs in most browsers.

Example code taken from linked article:

var typedArray = GetTheTypedArraySomehow();
var blob = new Blob([typedArray], {type: 'application/octet-binary'});
    // pass a useful mime type here
var url = URL.createObjectURL(blob);
// url will be something like: blob:d3958f5c-0777-0845-9dcf-2cb28783acaf
// now you can use the url in any context that regular URLs can be used
// in, for example img.src, etc.

Yes it is, you could do something like this in HTML5, using the download attribute

function download_txt() {
  var textToSave = document.getElementById('txt').innerHTML;
  var hiddenElement = document.createElement('a');

  hiddenElement.href = 'data:attachment/text,' + encodeURI(textToSave);
  hiddenElement.target = '_blank';
  hiddenElement.download = 'myFile.txt';
  hiddenElement.click();
}

document.getElementById('test').addEventListener('click', download_txt);

FIDDLE

To store a javascript variable, I suggest you to use libraries of data storage just like this one. where you can set a variable, get it , remove it ...

$.setData("key","value");
$.getData("key");
$.removeData("key");

but to store it on a file and make downloadable you have to pass by the server unless you use a javascript trick to download a file which doesn't exist, you just declare these functions

 var Download = 
    {
        click : function(node) {
            var ev = document.createEvent("MouseEvents");
            ev.initMouseEvent("click", true, false, self, 0, 0, 0, 0, 0, false, false, false, false, 0, null);
            return node.dispatchEvent(ev);
        },
        encode : function(data) {
                return 'data:application/octet-stream;base64,' + btoa( data );
        },
        link : function(data, name){
            var a = document.createElement('a');
            a.download = name || self.location.pathname.slice(self.location.pathname.lastIndexOf('/')+1);
            a.href = data || self.location.href;
            return a;
        }
    };
    Download.save = function(data, name)
    {
        this.click(
            this.link(
                this.encode( data ),
                name
            )
        );
    };

and when you want to download a file, you do this

Download.save("data to be on a file","FileName.txt");

Finally, you need to combine the datastorage and the filedownload solution to get the result you're asking for